Some Mathematical Methods for Neurosciences

Présentation du cours 2021-2022 (avec E. Tanré)

Master M2 UPMC

Master M2 MVA

Stages Master - Biblio. / Refs. - Cours / Lectures - Where / When 


We present a number of mathematical tools that are central to modeling in neuroscience. The prerequisites to the course are a good knowledge of differential calculus and probability theory from the viewpoint of measure theory. The thrust of the lectures is to show the applicability to neuroscience of the mathematical concepts without giving up mathematical rigor. The concepts presented in the lectures will be illustrated by exercise sessions.

  • Introduction to dynamical systems: orbits and phase portraits, invariant manifolds, center manifold in finite dimension.
  • Introduction to bifurcation theory: dimension 1 (saddle-node, transcritical, pitchfork), dimension 2 (Hopf), center manifold, normal form, equivariant bifurcations.
  • Applications: single spiking neuron dynamics, Turing mechanism for cortical pattern formation, geometric visual hallucinations.
  • Mesoscopic models of visual cortical areas: anatomical structure of the visual cortex (V1), functional architecture of V1, neural fields models.
  • Neuronal models: aspatial Hodgkin-Huxley model, simplified models, synaptic models, spatial models.
  • Importance of noise: Brownian motion, stochastic differential equations, application to neurons.

Résumé


Nous présentons dans ce cours quelques outils mathématiques qui interviennent de manière systématique dans de nombreux problèmes de modélisation en neurosciences. Les prérequis sont une bonne connaissance du calcul différentiel et du calcul des probabilités dans le cadre de la théorie de la mesure. Sans trahir la rigueur mathématique, le cours s'efforcera de mettre en valeur l'applicabilité aux neurosciences des concepts présentés. Le cours sera complété par des séances d'exercices.

  • Introduction aux systèmes dynamiques: orbites et portraits de phases, variétés invariantes, équivalence de systèmes dynamiques, classification topologique des équilibres, stabilité structurelle, variété centrale en dimension finie.
  • Introduction à la théorie des bifurcations: dimension 1 (noeud-selle, transcritique, fourche), dimension 2 (Hopf), variété centrale, forme normale, bifurcations équivariantes.

Applications:

  • Modèles mésoscopiques de certaines structures corticales: structure anatomique du cortex visuel (aire V1), architecture fonctionnelle de V1, modèles de champs neuronaux.
  • Sensibilité à l'orientation des contours visuels, formation de structures corticales et hallucinations visuelles.
  • Modèles de neurones: le modèle de Hodgkin-Huxley sans espace, modèles simpliés, modèles de synapses, modèles spatiaux.
  • Le rôle du bruit: mouvement Brownien, équations différentielles stochastiques, application aux neurones.

Interships Master


Here are the projects proposed this year (more to come):

Bibliographie sommaire (A few references)


  1. Kandel, Eric R., éd. Principles of neural science. 5th ed. New York: McGraw-Hill, 2013.
  2. Byrne, John H., Ruth Heidelberger, et Melvin Neal Waxham, From molecules to networks: an introduction to cellular and molecular neuroscience, 2014.
  3. Gerstner, Wulfram, Werner M. Kistler, Richard Naud, et Liam Paninski. Neuronal dynamics: from single neurons to networks and models of cognition. Cambridge University Press, 2014.
  4. Koch, Christof. Biophysics of Computation: Information Processing in Single Neurons. Oxford Univ. Press, 2004.
  5. Bressloff, Paul C. Waves in Neural Media, Springer, 2014.
  6. Eugène Izhikevich, Dynamical systems in neuroscience: the geometry of excitability and bursting, MIT Press, 2006.
  7. G. Bard Ermentrout and David H. Terman, Mathematical Foundations of Neuroscience, Springer, 2010.
  8. Sterratt, David, Principles of computational modelling in neuroscience. Cambridge University Press, 2011.

  1. Haragus, Mariana, et Gerard Iooss. Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems. London: Springer London, 2011. http://link.springer.com/10.1007/978-0-85729-112-7.
  2. Meiss, Differential Dynamical Systems. SIAM, 2007
  3. Sylvie Benzoni, Cours de M1 sur les EDOs, [Link]
  4. Yuri A. Kuznetsov, Elements of applied bifurcation theory.
  5. Jean-Pierre Françoise, Oscillations en biologie, Springer, 2000.
  6. Lawrence C. Evans, An introduction to stochastic differential equations, [Link]
  7. Jean-François Le Gall, Mouvement brownien, martingales et calcul stochastique, 2013. [Link]

Date et lieu des cours et des TPs ( When and where)


Location: Campus Jussieu : couloir 14/24, room 108

Les cours ont lieu les jeudis de 13:30 à 16:30, les séances de TDs de 16h45-18h45

The lectures will be given at Jussieu from 13:30 to 16:30, tutorials: 16h45-18h45.

October, 21st (slides)


Location: Campus Jussieu : couloir 14/24, salle 108.

This document contains basic results useful for the lectures. One may also look at Meiss, Differential Dynamical Systems. SIAM, 2007.

  • Introduction to the Central Nervous System CNS
  • Models of a single neuron ( Hodgkin-Huxley ) 
  • Basics of dynamical systems (existence theorem, stability)
  • Introduction to planar models of single neurons (Morris–Lecar , FitzHugh-Nagumo, Integrate and Fire, Exponential Integrate and Fire...)

Exercises

October, 28th (slides)


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Location: Campus Jussieu : couloir 14/24, salle 108.

  • Introduction to noise in models in neuroscience
  • Homogeneous Poisson processes
  • Inhomogeneous Poisson processes
  • Exact simulation of inhomogeneous Poisson processes (link with Point Poisson Processes in R^2)

November, 4th


Location: Campus ENS Paris Saclay Bat Nord Salle 1Z28

  • Point Poisson Processes
  • Continuous Time Markov Processes

November, 18th (Slides)


Location: Campus Jussieu : couloir 14/24, salle 108.

  • Introduction to local bifurcation theory (codim 1)
  • Examples of bifurcations in neural models of single point neuron

Exercises solutions

Optional exercise about Existence of unstable manifold.

November, 25th (Slides)


Location: Campus Jussieu : couloir 14/24, salle 108.

  • Synaptic transmission
  • Normal form theory
  • Delayed Differential Equations
  • Applications to models of neural networks

Exercises

2 December (Slides)


Location: Campus Jussieu : couloir 14/24, salle 108.

Lecture notes summing up most notions reviewed in this course.

  • Normal form theory
  • Models of visual cortex and their analysis

Exercises

9 December 


Location: Campus Jussieu : couloir 14/24, salle 108.

  • Piecewise-deterministic Markov Processes. Examples in neuroscience.

16 December
(Slides Landscape format and Slides Portrait format)


Location: Campus Jussieu : couloir 14/24, salle 108.

Examen January 6th 2022


Location: Campus Jussieu salle 54/55-206.

Time: 9h00-12h00

Room: 54/55-206.

4 pages of written notes are allowed. 

Previous exams

Previous exams